Higher spin multiplicity means lower energy. I.e.: you want to keep all spins pointin in the same direction.

Can we make any ab initio predictions about it all?

A 2016 paper: aip.scitation.org/doi/abs/10.1063/1.4948309

Isomers were quite confusing for early chemists, before atomic theory was widely accepted, and people where thinking mostly in terms of proportions of equations, related: Section "Isomers suggest that atoms exist".

Exist because double bonds don't rotate freely. Have different properties of course, unlike enantiomer.

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Molecules that are the same if you just look at "what atom is linked to what atom", they are only different if you consider the relative spacial positions of atoms.

A more concrete and easier to understand version of it is the more photon-specific Poincaré sphere, have a look at that one first.

One dimension in position representation:

In three dimensions In position representation, we define it by using the gradient, and so we see that

Appears directly on Schrödinger equation! And in particular in the time-independent Schrödinger equation.

Basically the operators are just analogous to the classical ones e.g. the classical:becomes:

Besides the angular momentum in each direction, we also have the total angular momentum:

Then you have to understand what each one of those does to the each atomic orbital:

There is an uncertainty principle between the x, y and z angular momentums, we can only measure one of them with certainty at a time. Video 1. "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)" justifies this intuitively by mentioning that this is analogous to precession: if you try to measure electrons e.g. with the Zeeman effect the precess on the other directions which you end up modifing.

TODO experiment. Likely Zeeman effect.

See: angular momentum operator.

Proof that the probability 1 is conserved by the time evolution:

It can be derived directly from the Schrödinger equation.

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Contains the full state of the quantum system.

This is in contrast to classical mechanics where e.g. the state of mechanical system is given by two real functions: position and speed.

The wave equation in position representation on the other hand encodes speed in "how fast does the complex phase spin around", and direction in "does it spin clockwise or counterclockwise", as described well at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)". Then once you understand that, it is more compact to just view those graphs with the phase color coded as in Video "Simulation of the time-dependent Schrodinger equation (JavaScript Animation) by Coding Physics (2019)".

Basically the same as matrix mechanics it seems, just a bit more generalized.

Photon energy is proportional to its frequency:or with common weird variables:

This only makes sense if the photon exists, there is no classical analogue, because the energy of classical waves depends only on their amplitude, not frequency.

Experiments that suggest this:

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